Abstract
In this introductory paper we take partial stock of the current state of field on calculus research, exemplifying both the promise of research advances as well as the limitations. We identify four trends in the calculus research literature, starting with identifying misconceptions to investigations of the processes by which students learn particular concepts, evolving into classroom studies, and, more recently research on teacher knowledge, beliefs, and practices. These trends are related to a model for the cycle of research and development aimed at improving learning and teaching. We then make use of these four trends and the model for the cycle of research and development to highlight the contributions of the papers in this issue. We conclude with some reflections on the gaps in literature and what new areas of calculus research are needed.
Similar content being viewed by others
References
Alves, F. R. V. (2012). Exploração de noções topológicas na transição do Cálculo para a Análise Real com o GeoGebra. Revista Do Instituto Geogebra Internacional de São Paulo, 1, 165–179.
Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Fuentes, S., Trigueros, M., et al. (2014). APOS theory: A framework for research and curriculum development in mathematics education. New York, NY: Springer.
Artigue, M. (1994). Didactical engineering as a framework for the conception of teaching products. In R. Biehler, R. W. Scholz, R. Sträßer, & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 27–39). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Artigue, M., & Mariotti, M. A. (2014). Networking theoretical frames: the ReMath enterprise. Educational Studies in Mathematics, 85, 329–355.
Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. E. (1997). Networking theoretical frames: the development of students’ graphical understanding of the derivative. The Journal of Mathematical Behavior, 16(4), 399–431.
Aspinwall, L., Shaw, K. L., & Presmeg, N. C. (1997). Uncontrollable mental imagery: graphical connections between a function and its derivative. Educational Studies in Mathematics, 33(3), 301–317.
Baldino, R. R., & Cabral, T. C. B. (1994). Os quatro discursos de Lacan eo Teorema Fundamental do Cálculo. Revista Quadrante, pp. 1–24.
Barbosa, S.M. (2009). Tecnologias da Informação e Comunicação, Função Composta e Regra da Cadeia. Doctoral Dissertation, UNESP, Rio Claro, Brazil.
Bergé, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis. Educational Studies in Mathematics, 67, 217–235.
Bikner-Ahsbahs, A., & Prediger, S. (2014) (Eds.), Networking of theories as a research practice. New York, NY: Springer.
Borba, M. C., & Villarreal, M. E. (2005). Humans-with-media and the reorganization of mathematical thinking: Information and communication technologies, modeling, experimentation and visualization. New York, NY: Springer.
Britton, S., & Henderson, J. (2013) Issues and trends: a review of Delta conference papers from 1997 to 2011. Lighthouse Delta 2013: The 9th Delta Conference on teaching and learning of undergraduate mathematics and statistics, pp. 50–58.
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221–266.
Christensen, W. M., & Thompson, J. R. (2012). Investigating graphical representations of slope and derivative without a physics context. Physical Review Special Topics-Physics Education Research, 8(2). http://journals.aps.org/prstper/abstract/10.1103/PhysRevSTPER.8.023101.
Cobb, P. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1), 2–33.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schaubel, L. (2006). Design experiments in educational research. Educational Researcher, 32(1), 99–113.
Code, W., Piccolo, C., Kohler, D., & MacLean, M. (2014, this issue). Teaching methods comparison in a large calculus class. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0582-2.
Davis, R. B., & Vinner, S. (1986). The notion of limit: some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5(3), 281–303.
diSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. Pufall (Eds.), Constructivism in the computer age (pp. 49–70). Hillsdale, NJ: Lawrence Erlbaum Associates.
Dullius, M. M., Araujo, I. S., & Veit, E. A. (2011). Ensino e Aprendizagem de Equações Diferenciais com Abordagem Gráfica, Numérica e Analítica: um experiência em Cursos de Engenharia. Bolema. Boletim de Educação Matemática (UNESP. Rio Claro. Impresso), Vol. 24, pp. 17–42.
Eichler, A., & Erens, R. (2014, this issue). Teachers’ beliefs towards teaching calculus. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0606-y.
Ellis, J., Kelton, M. L., & Rasmussen, C. (2014, this issue). Student perceptions of pedagogy and associated persistence in calculus. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0577-z.
Ferrini-Mundy, J., & Graham, K. (1991). An overview of the calculus curriculum reform effort: issues for learning, teaching, and curriculum development. American Mathematical Monthly, 98(7), 627–635.
Furinghetti, F., & Paola, D. (1988). Wrong beliefs and misunderstandings about basic concepts of calculus (age 16–19). In Proceedings of the 39th rencontre internationale de la CIEAEM, 1987 (pp. 173-177). Sherbrooke, Canada.
Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht, The Netherlands: CD Press.
Gray, E., & Tall, D. (1994). Duality, ambiguity, and flexibility: a “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140.
Hähkiöniemi, M. (2004). Perceptual and symbolic representations as a starting point of the acquisition of the derivative. In Hoines, M. J. & Fuglestad, A. B. (Eds.), Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 73–80.
Hannula, M. S. (2012). Exploring new dimensions of mathematics-related affect: embodied and social theories. Research in Mathematics Education, 14(2), 137–161.
Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: epistemic actions. Journal for Research in Mathematics Education, 32, 195–222.
Hershkowitz, R., Tabach, M., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts in a probability class: a case study. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0576-0.
Javaroni, S. (2007). Abordagem geométrica: possibilidades para o ensino e aprendizagem de Introdução às Equações Diferenciais Ordinária, Doctoral Disssertation. Rio Claro, Brazil: UNESP.
Job, P., & Schneider, M. (2014, this issue). Empirical positivism, an epistemological obstacle in the learning of calculus. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0604-0.
Kabael, T. (2010). Cognitive development of applying the chain rule through three worlds of mathematics. Australian Senior Mathematics Journal, 24(2), 14–28.
Keene, K. (2007). A characterization of dynamic reasoning: reasoning with time as parameter. Journal of Mathematical Behavior, 26, 230–246.
Keene, K. A., Hall, W., & Duca, A. (2014, this issue). Sequence limits in calculus: using design research and building on intuition to support instruction. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0597-8.
Kelly, A. E., Lesh, R. A., & Baek, J. Y. (Eds.). (2008). Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching. London: Routledge.
Kouropatov, A., & Dreyfus, T. (2014, this issue). Learning the integral concept by constructing knowledge about accumulation. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0571-5.
Larsen, S., Johnson, E., & Weber, K. (2013). (Eds.) The teaching abstract algebra for understanding project: designing and scaling up a curriculum innovation. Journal of Mathematical Behavior, 32(4).
Lesh, R., & Kelly, A. (2000). Multi-tiered teaching experiments. In A. Kelly & R. Lesh (Eds.), Handbook of research in mathematics and science education (pp. 197–230). Mahwah, NJ: Lawrence Erlbaum.
Marrongelle, K. (2004). Context, examples, and language: students uses of physics to reason about calculus. School Science and Mathematics, 10(6), 258–272.
Moreno-Armella, L. (2014, this issue). An essential tension in mathematics education. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0580-4.
Morgan, R. V., & Warnock, T. T. (1978). Derivatives on the hand-held calculator. Mathematics Teacher, 71(6), 532–537.
Nemirovsky, R., & Rubin, A. (1992). Students’ tendency to assume resemblances between a function and its derivative, TERC Working Paper 2–92, Cambridge.
Orton, A. A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14(3), 235–250.
Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework. ZDM—The International Journal on Mathematics Education, 40(2), 165–178.
Radford, L. (2003). Gestures, speech, and the sprouting of signs: a semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.
RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: Toward a strategic research program in mathematics education. Santa Monica, CA: RAND Corporation.
Rasmussen, C. (Ed.) (2007). An inquiry oriented approach to differential equations. Journal of Mathematical Behavior, 26(3) (special issue).
Rasmussen, C., Wawro, M., & Zandieh, M. (2012). Four lenses for examining individual and collective level mathematical progress. Paper presented at the annual meeting of the American Educational Research Association, Vancouver, Canada.
Salinas, P. (2013). Approaching calculus with SimCalc: linking derivative and antiderivative. In S. J. Hegedus & J. Roschelle (Eds.), The SimCalc vision and contributions (pp. 383–399). Dordrecht, The Netherlands: Springer.
Schoenfeld, A. (1994). Some notes on the enterprise (research in collegiate mathematics education, that is). Research in Collegiate Mathematics Education, 1, 1–19.
Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245.
Sfard, A. (1991a). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.
Sfard, A. (1991b). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.
Shadish, W. R., Cook, T. D., & Campbell, D. T. (2001). Experimental and quasi-experimental designs for generalized causal inference. Boston: Houghton Mifflin.
Soares, D. S. (2012). Uma abordagem pedagógica baseada na análise de modelos para alunos de biologia: qual o papel do software? Doctoral Dissertation, UNESP, Brazil.
Soares, D. S., & Borba, M. (2014, this issue). The role of software Modellus in a teaching approach based on model analysis. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-013-0568-5.
Swidan, O., & Yerushalmy, M. (2014, this issue). Learning the indefinite integral in a dynamic and interactive technological environment. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0583-1.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.
Thompson, P. W. (1992). Notations, conventions, and constraints: contributions to effective uses of concrete materials in elementary mathematics. Journal for Research in Mathematics Education, 23, 123–147.
Thompson, P. W., & Silverman, J. (2008). The concept of accumulation in calculus. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 117–131). Washington, DC: The Mathematical Association of America.
Tinto, V. (2004). Linking learning and leaving. In J. M. Braxton (Ed.), Reworking the student departure puzzle. Nashville, TN: Vanderbilt University Press.
Törner, G., Potari, D., & Zachariades, T. (2014, this issue). Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0612-0.
Trigueros, M., & Martínez-Planell, R. (2010). Geometrical representations in the learning of two-variable functions. Educational Studies in Mathematics, 73(1), 3–19.
Weigand, H.-G. (2014, this issue). A discrete approach to the concept of derivative. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0595-x.
White, N., & Mesa, V. (2014, this issue). Describing cognitive orientation of Calculus I tasks across different types of coursework. ZDM—The International Journal on Mathematics Education. doi:10.1007/s11858-014-0588-9.
Yoon, C., Thomas, M. J., & Dreyfus, T. (2011a). Grounded blends and mathematical gesture spaces: developing mathematical understandings via gestures. Educational Studies in Mathematics, 78(3), 371–393.
Yoon, C., Thomas, M. J., & Dreyfus, T. (2011b). Gestures and insight in advanced mathematical thinking. International Journal of Mathematical Education in Science and Technology, 42(7), 891–901.
Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. Research in Collegiate Mathematics Education. IV, 103–127.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rasmussen, C., Marrongelle, K. & Borba, M.C. Research on calculus: what do we know and where do we need to go?. ZDM Mathematics Education 46, 507–515 (2014). https://doi.org/10.1007/s11858-014-0615-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11858-014-0615-x